metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊7Q8, C42.146D6, C6.942- 1+4, C4.15(S3×Q8), C4⋊C4.202D6, C3⋊4(Q8⋊3Q8), C12⋊Q8.12C2, C12.47(C2×Q8), C42.C2.6S3, C6.39(C22×Q8), (C2×C6).229C24, (C2×C12).84C23, C2.55(Q8○D12), (C4×Dic6).24C2, Dic3.13(C2×Q8), Dic3.Q8.2C2, (C4×C12).190C22, C4.Dic6.12C2, Dic3.40(C4○D4), Dic6⋊C4.11C2, C4⋊Dic3.378C22, C22.250(S3×C23), Dic3⋊C4.143C22, (C2×Dic6).298C22, (C2×Dic3).119C23, (C4×Dic3).137C22, C2.22(C2×S3×Q8), C2.82(S3×C4○D4), C6.193(C2×C4○D4), (C2×C4).75(C22×S3), (C3×C42.C2).5C2, (C3×C4⋊C4).184C22, SmallGroup(192,1244)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊7Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=dbd-1=a6b, dcd-1=c-1 >
Subgroups: 400 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×Q8, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×Q8, C42.C2, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, Q8⋊3Q8, C4×Dic6, Dic6⋊C4, Dic6⋊C4, C12⋊Q8, C12⋊Q8, Dic3.Q8, C4.Dic6, C3×C42.C2, Dic6⋊7Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, 2- 1+4, S3×Q8, S3×C23, Q8⋊3Q8, C2×S3×Q8, S3×C4○D4, Q8○D12, Dic6⋊7Q8
►Regular action on 192 points
Generators in S192
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 181 7 187)(2 192 8 186)(3 191 9 185)(4 190 10 184)(5 189 11 183)(6 188 12 182)(13 168 19 162)(14 167 20 161)(15 166 21 160)(16 165 22 159)(17 164 23 158)(18 163 24 157)(25 110 31 116)(26 109 32 115)(27 120 33 114)(28 119 34 113)(29 118 35 112)(30 117 36 111)(37 155 43 149)(38 154 44 148)(39 153 45 147)(40 152 46 146)(41 151 47 145)(42 150 48 156)(49 64 55 70)(50 63 56 69)(51 62 57 68)(52 61 58 67)(53 72 59 66)(54 71 60 65)(73 127 79 121)(74 126 80 132)(75 125 81 131)(76 124 82 130)(77 123 83 129)(78 122 84 128)(85 173 91 179)(86 172 92 178)(87 171 93 177)(88 170 94 176)(89 169 95 175)(90 180 96 174)(97 133 103 139)(98 144 104 138)(99 143 105 137)(100 142 106 136)(101 141 107 135)(102 140 108 134)
(1 40 77 61)(2 41 78 62)(3 42 79 63)(4 43 80 64)(5 44 81 65)(6 45 82 66)(7 46 83 67)(8 47 84 68)(9 48 73 69)(10 37 74 70)(11 38 75 71)(12 39 76 72)(13 88 104 35)(14 89 105 36)(15 90 106 25)(16 91 107 26)(17 92 108 27)(18 93 97 28)(19 94 98 29)(20 95 99 30)(21 96 100 31)(22 85 101 32)(23 86 102 33)(24 87 103 34)(49 190 155 132)(50 191 156 121)(51 192 145 122)(52 181 146 123)(53 182 147 124)(54 183 148 125)(55 184 149 126)(56 185 150 127)(57 186 151 128)(58 187 152 129)(59 188 153 130)(60 189 154 131)(109 159 179 141)(110 160 180 142)(111 161 169 143)(112 162 170 144)(113 163 171 133)(114 164 172 134)(115 165 173 135)(116 166 174 136)(117 167 175 137)(118 168 176 138)(119 157 177 139)(120 158 178 140)
(1 172 77 114)(2 177 78 119)(3 170 79 112)(4 175 80 117)(5 180 81 110)(6 173 82 115)(7 178 83 120)(8 171 84 113)(9 176 73 118)(10 169 74 111)(11 174 75 116)(12 179 76 109)(13 50 104 156)(14 55 105 149)(15 60 106 154)(16 53 107 147)(17 58 108 152)(18 51 97 145)(19 56 98 150)(20 49 99 155)(21 54 100 148)(22 59 101 153)(23 52 102 146)(24 57 103 151)(25 189 90 131)(26 182 91 124)(27 187 92 129)(28 192 93 122)(29 185 94 127)(30 190 95 132)(31 183 96 125)(32 188 85 130)(33 181 86 123)(34 186 87 128)(35 191 88 121)(36 184 89 126)(37 161 70 143)(38 166 71 136)(39 159 72 141)(40 164 61 134)(41 157 62 139)(42 162 63 144)(43 167 64 137)(44 160 65 142)(45 165 66 135)(46 158 67 140)(47 163 68 133)(48 168 69 138)
G:=sub<Sym(192)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,181,7,187)(2,192,8,186)(3,191,9,185)(4,190,10,184)(5,189,11,183)(6,188,12,182)(13,168,19,162)(14,167,20,161)(15,166,21,160)(16,165,22,159)(17,164,23,158)(18,163,24,157)(25,110,31,116)(26,109,32,115)(27,120,33,114)(28,119,34,113)(29,118,35,112)(30,117,36,111)(37,155,43,149)(38,154,44,148)(39,153,45,147)(40,152,46,146)(41,151,47,145)(42,150,48,156)(49,64,55,70)(50,63,56,69)(51,62,57,68)(52,61,58,67)(53,72,59,66)(54,71,60,65)(73,127,79,121)(74,126,80,132)(75,125,81,131)(76,124,82,130)(77,123,83,129)(78,122,84,128)(85,173,91,179)(86,172,92,178)(87,171,93,177)(88,170,94,176)(89,169,95,175)(90,180,96,174)(97,133,103,139)(98,144,104,138)(99,143,105,137)(100,142,106,136)(101,141,107,135)(102,140,108,134), (1,40,77,61)(2,41,78,62)(3,42,79,63)(4,43,80,64)(5,44,81,65)(6,45,82,66)(7,46,83,67)(8,47,84,68)(9,48,73,69)(10,37,74,70)(11,38,75,71)(12,39,76,72)(13,88,104,35)(14,89,105,36)(15,90,106,25)(16,91,107,26)(17,92,108,27)(18,93,97,28)(19,94,98,29)(20,95,99,30)(21,96,100,31)(22,85,101,32)(23,86,102,33)(24,87,103,34)(49,190,155,132)(50,191,156,121)(51,192,145,122)(52,181,146,123)(53,182,147,124)(54,183,148,125)(55,184,149,126)(56,185,150,127)(57,186,151,128)(58,187,152,129)(59,188,153,130)(60,189,154,131)(109,159,179,141)(110,160,180,142)(111,161,169,143)(112,162,170,144)(113,163,171,133)(114,164,172,134)(115,165,173,135)(116,166,174,136)(117,167,175,137)(118,168,176,138)(119,157,177,139)(120,158,178,140), (1,172,77,114)(2,177,78,119)(3,170,79,112)(4,175,80,117)(5,180,81,110)(6,173,82,115)(7,178,83,120)(8,171,84,113)(9,176,73,118)(10,169,74,111)(11,174,75,116)(12,179,76,109)(13,50,104,156)(14,55,105,149)(15,60,106,154)(16,53,107,147)(17,58,108,152)(18,51,97,145)(19,56,98,150)(20,49,99,155)(21,54,100,148)(22,59,101,153)(23,52,102,146)(24,57,103,151)(25,189,90,131)(26,182,91,124)(27,187,92,129)(28,192,93,122)(29,185,94,127)(30,190,95,132)(31,183,96,125)(32,188,85,130)(33,181,86,123)(34,186,87,128)(35,191,88,121)(36,184,89,126)(37,161,70,143)(38,166,71,136)(39,159,72,141)(40,164,61,134)(41,157,62,139)(42,162,63,144)(43,167,64,137)(44,160,65,142)(45,165,66,135)(46,158,67,140)(47,163,68,133)(48,168,69,138)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,181,7,187)(2,192,8,186)(3,191,9,185)(4,190,10,184)(5,189,11,183)(6,188,12,182)(13,168,19,162)(14,167,20,161)(15,166,21,160)(16,165,22,159)(17,164,23,158)(18,163,24,157)(25,110,31,116)(26,109,32,115)(27,120,33,114)(28,119,34,113)(29,118,35,112)(30,117,36,111)(37,155,43,149)(38,154,44,148)(39,153,45,147)(40,152,46,146)(41,151,47,145)(42,150,48,156)(49,64,55,70)(50,63,56,69)(51,62,57,68)(52,61,58,67)(53,72,59,66)(54,71,60,65)(73,127,79,121)(74,126,80,132)(75,125,81,131)(76,124,82,130)(77,123,83,129)(78,122,84,128)(85,173,91,179)(86,172,92,178)(87,171,93,177)(88,170,94,176)(89,169,95,175)(90,180,96,174)(97,133,103,139)(98,144,104,138)(99,143,105,137)(100,142,106,136)(101,141,107,135)(102,140,108,134), (1,40,77,61)(2,41,78,62)(3,42,79,63)(4,43,80,64)(5,44,81,65)(6,45,82,66)(7,46,83,67)(8,47,84,68)(9,48,73,69)(10,37,74,70)(11,38,75,71)(12,39,76,72)(13,88,104,35)(14,89,105,36)(15,90,106,25)(16,91,107,26)(17,92,108,27)(18,93,97,28)(19,94,98,29)(20,95,99,30)(21,96,100,31)(22,85,101,32)(23,86,102,33)(24,87,103,34)(49,190,155,132)(50,191,156,121)(51,192,145,122)(52,181,146,123)(53,182,147,124)(54,183,148,125)(55,184,149,126)(56,185,150,127)(57,186,151,128)(58,187,152,129)(59,188,153,130)(60,189,154,131)(109,159,179,141)(110,160,180,142)(111,161,169,143)(112,162,170,144)(113,163,171,133)(114,164,172,134)(115,165,173,135)(116,166,174,136)(117,167,175,137)(118,168,176,138)(119,157,177,139)(120,158,178,140), (1,172,77,114)(2,177,78,119)(3,170,79,112)(4,175,80,117)(5,180,81,110)(6,173,82,115)(7,178,83,120)(8,171,84,113)(9,176,73,118)(10,169,74,111)(11,174,75,116)(12,179,76,109)(13,50,104,156)(14,55,105,149)(15,60,106,154)(16,53,107,147)(17,58,108,152)(18,51,97,145)(19,56,98,150)(20,49,99,155)(21,54,100,148)(22,59,101,153)(23,52,102,146)(24,57,103,151)(25,189,90,131)(26,182,91,124)(27,187,92,129)(28,192,93,122)(29,185,94,127)(30,190,95,132)(31,183,96,125)(32,188,85,130)(33,181,86,123)(34,186,87,128)(35,191,88,121)(36,184,89,126)(37,161,70,143)(38,166,71,136)(39,159,72,141)(40,164,61,134)(41,157,62,139)(42,162,63,144)(43,167,64,137)(44,160,65,142)(45,165,66,135)(46,158,67,140)(47,163,68,133)(48,168,69,138) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,181,7,187),(2,192,8,186),(3,191,9,185),(4,190,10,184),(5,189,11,183),(6,188,12,182),(13,168,19,162),(14,167,20,161),(15,166,21,160),(16,165,22,159),(17,164,23,158),(18,163,24,157),(25,110,31,116),(26,109,32,115),(27,120,33,114),(28,119,34,113),(29,118,35,112),(30,117,36,111),(37,155,43,149),(38,154,44,148),(39,153,45,147),(40,152,46,146),(41,151,47,145),(42,150,48,156),(49,64,55,70),(50,63,56,69),(51,62,57,68),(52,61,58,67),(53,72,59,66),(54,71,60,65),(73,127,79,121),(74,126,80,132),(75,125,81,131),(76,124,82,130),(77,123,83,129),(78,122,84,128),(85,173,91,179),(86,172,92,178),(87,171,93,177),(88,170,94,176),(89,169,95,175),(90,180,96,174),(97,133,103,139),(98,144,104,138),(99,143,105,137),(100,142,106,136),(101,141,107,135),(102,140,108,134)], [(1,40,77,61),(2,41,78,62),(3,42,79,63),(4,43,80,64),(5,44,81,65),(6,45,82,66),(7,46,83,67),(8,47,84,68),(9,48,73,69),(10,37,74,70),(11,38,75,71),(12,39,76,72),(13,88,104,35),(14,89,105,36),(15,90,106,25),(16,91,107,26),(17,92,108,27),(18,93,97,28),(19,94,98,29),(20,95,99,30),(21,96,100,31),(22,85,101,32),(23,86,102,33),(24,87,103,34),(49,190,155,132),(50,191,156,121),(51,192,145,122),(52,181,146,123),(53,182,147,124),(54,183,148,125),(55,184,149,126),(56,185,150,127),(57,186,151,128),(58,187,152,129),(59,188,153,130),(60,189,154,131),(109,159,179,141),(110,160,180,142),(111,161,169,143),(112,162,170,144),(113,163,171,133),(114,164,172,134),(115,165,173,135),(116,166,174,136),(117,167,175,137),(118,168,176,138),(119,157,177,139),(120,158,178,140)], [(1,172,77,114),(2,177,78,119),(3,170,79,112),(4,175,80,117),(5,180,81,110),(6,173,82,115),(7,178,83,120),(8,171,84,113),(9,176,73,118),(10,169,74,111),(11,174,75,116),(12,179,76,109),(13,50,104,156),(14,55,105,149),(15,60,106,154),(16,53,107,147),(17,58,108,152),(18,51,97,145),(19,56,98,150),(20,49,99,155),(21,54,100,148),(22,59,101,153),(23,52,102,146),(24,57,103,151),(25,189,90,131),(26,182,91,124),(27,187,92,129),(28,192,93,122),(29,185,94,127),(30,190,95,132),(31,183,96,125),(32,188,85,130),(33,181,86,123),(34,186,87,128),(35,191,88,121),(36,184,89,126),(37,161,70,143),(38,166,71,136),(39,159,72,141),(40,164,61,134),(41,157,62,139),(42,162,63,144),(43,167,64,137),(44,160,65,142),(45,165,66,135),(46,158,67,140),(47,163,68,133),(48,168,69,138)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | 2- 1+4 | S3×Q8 | S3×C4○D4 | Q8○D12 |
| kernel | Dic6⋊7Q8 | C4×Dic6 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | C3×C42.C2 | C42.C2 | Dic6 | C42 | C4⋊C4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 4 | 3 | 4 | 1 | 1 | 1 | 4 | 1 | 6 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6⋊7Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
Dic6⋊7Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_7Q_8 % in TeX
G:=Group("Dic6:7Q8"); // GroupNames label
G:=SmallGroup(192,1244);
// by ID
G=gap.SmallGroup(192,1244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations